Dear GAP - Forum,

M. Lavrauw wrote:

To find a primitive polynomial over a finite field,
I did the following:

pring:=UnivariatePolynomialRing(<finite field>);
X:=IndeterminatesOfPolynomialRing(pring)[1];
SetName(X,"X");

Suppose the field is GF(q) and I want a primitive polynomial of degree 3.
Then I generate a random polynomial by generating random coefficients
in the field. Let f be such a random polynomial. To check if it is primitive
I make a list of all divisors d of q^3-1 such that (q^3-1)/x is prime.
Then f is primitive if X^d mod f is different from One(GF(q)) for all
d in list with d < q^3-1.

Calculating X^d mod f takes a long time in Gap ( even if we work over a
prime field).

May it be that you calculate X^d before reducing it modulo f ?
This would certainly take a very long time (and also much memory) if
q is 'large'.
Calculating X^d mod f could be done much faster by a simple method
analogous to that used for modular exponentiation for integers, which
reduces the result after each multiplication modulo the respective
modulus.

Best regards,

Stefan Kohl